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-rw-r--r--theories/Arith/Minus.v32
1 files changed, 16 insertions, 16 deletions
diff --git a/theories/Arith/Minus.v b/theories/Arith/Minus.v
index ed215f54..48024331 100644
--- a/theories/Arith/Minus.v
+++ b/theories/Arith/Minus.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -21,7 +21,7 @@ where "n - m" := (minus n m) : nat_scope.
Require Import Lt.
Require Import Le.
-Open Local Scope nat_scope.
+Local Open Scope nat_scope.
Implicit Types m n p : nat.
@@ -29,7 +29,7 @@ Implicit Types m n p : nat.
Lemma minus_n_O : forall n, n = n - 0.
Proof.
- induction n; simpl in |- *; auto with arith.
+ induction n; simpl; auto with arith.
Qed.
Hint Resolve minus_n_O: arith v62.
@@ -37,21 +37,21 @@ Hint Resolve minus_n_O: arith v62.
Lemma minus_Sn_m : forall n m, m <= n -> S (n - m) = S n - m.
Proof.
- intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *;
+ intros n m Le; pattern m, n; apply le_elim_rel; simpl;
auto with arith.
Qed.
Hint Resolve minus_Sn_m: arith v62.
Theorem pred_of_minus : forall n, pred n = n - 1.
Proof.
- intro x; induction x; simpl in |- *; auto with arith.
+ intro x; induction x; simpl; auto with arith.
Qed.
(** * Diagonal *)
Lemma minus_diag : forall n, n - n = 0.
Proof.
- induction n; simpl in |- *; auto with arith.
+ induction n; simpl; auto with arith.
Qed.
Lemma minus_diag_reverse : forall n, 0 = n - n.
@@ -66,7 +66,7 @@ Notation minus_n_n := minus_diag_reverse.
Lemma minus_plus_simpl_l_reverse : forall n m p, n - m = p + n - (p + m).
Proof.
- induction p; simpl in |- *; auto with arith.
+ induction p; simpl; auto with arith.
Qed.
Hint Resolve minus_plus_simpl_l_reverse: arith v62.
@@ -74,7 +74,7 @@ Hint Resolve minus_plus_simpl_l_reverse: arith v62.
Lemma plus_minus : forall n m p, n = m + p -> p = n - m.
Proof.
- intros n m p; pattern m, n in |- *; apply nat_double_ind; simpl in |- *;
+ intros n m p; pattern m, n; apply nat_double_ind; simpl;
intros.
replace (n0 - 0) with n0; auto with arith.
absurd (0 = S (n0 + p)); auto with arith.
@@ -83,20 +83,20 @@ Qed.
Hint Immediate plus_minus: arith v62.
Lemma minus_plus : forall n m, n + m - n = m.
- symmetry in |- *; auto with arith.
+ symmetry ; auto with arith.
Qed.
Hint Resolve minus_plus: arith v62.
Lemma le_plus_minus : forall n m, n <= m -> m = n + (m - n).
Proof.
- intros n m Le; pattern n, m in |- *; apply le_elim_rel; simpl in |- *;
+ intros n m Le; pattern n, m; apply le_elim_rel; simpl;
auto with arith.
Qed.
Hint Resolve le_plus_minus: arith v62.
Lemma le_plus_minus_r : forall n m, n <= m -> n + (m - n) = m.
Proof.
- symmetry in |- *; auto with arith.
+ symmetry ; auto with arith.
Qed.
Hint Resolve le_plus_minus_r: arith v62.
@@ -132,7 +132,7 @@ Qed.
Lemma lt_minus : forall n m, m <= n -> 0 < m -> n - m < n.
Proof.
- intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *;
+ intros n m Le; pattern m, n; apply le_elim_rel; simpl;
auto using le_minus with arith.
intros; absurd (0 < 0); auto with arith.
Qed.
@@ -140,7 +140,7 @@ Hint Resolve lt_minus: arith v62.
Lemma lt_O_minus_lt : forall n m, 0 < n - m -> m < n.
Proof.
- intros n m; pattern n, m in |- *; apply nat_double_ind; simpl in |- *;
+ intros n m; pattern n, m; apply nat_double_ind; simpl;
auto with arith.
intros; absurd (0 < 0); trivial with arith.
Qed.
@@ -148,9 +148,9 @@ Hint Immediate lt_O_minus_lt: arith v62.
Theorem not_le_minus_0 : forall n m, ~ m <= n -> n - m = 0.
Proof.
- intros y x; pattern y, x in |- *; apply nat_double_ind;
- [ simpl in |- *; trivial with arith
+ intros y x; pattern y, x; apply nat_double_ind;
+ [ simpl; trivial with arith
| intros n H; absurd (0 <= S n); [ assumption | apply le_O_n ]
- | simpl in |- *; intros n m H1 H2; apply H1; unfold not in |- *; intros H3;
+ | simpl; intros n m H1 H2; apply H1; unfold not; intros H3;
apply H2; apply le_n_S; assumption ].
Qed.